1

So there's a word problem I found in my text that I've been trying to solve for about 2 hours now, and I just can't seem to get it.

A school has three clubs and each student is required to belong to exactly one club. One year the students switch club membership as follows.

Club A. $\frac{4}{10}$ remian in A, $\frac{1}{10}$ switch to B and $\frac{5}{10}$ switch to C.

Club B. $\frac{7}{10}$ remain in B, $\frac{2}{10}$ switch to A, and $\frac{1}{10}$ switch to C.

Club C. $\frac{6}{10}$ remian in C. $\frac{2}{10}$ switch to A, and $\frac{2}{10}$ switch to B

I came up with 3 equations:

$\frac{4}{10}A + \frac{2}{10}B + \frac{2}{10}c$ which should give the number of people in A.

$\frac{1}{10}A + \frac{7}{10}B + \frac{2}{10}c$ which should give the number of people in B

$\frac{5}{10}A + \frac{1}{10}B + \frac{6}{10}c$ which should give the number of people in C.

EDIT: If the fraction of the student population is unchanged, find each of these fractions."

Problem is, is that I don't know what each equation is equal to since we're not given the number of people in each club.

Thanks.

Community
  • 1
  • 2
    Your equations are right, but the problem as quoted doesn't seem to ask a question. What are you being asked to calculate? – Stella Biderman Sep 13 '17 at 18:41
  • If we call your matrix $T$, then we have the very cool equation $v_f=Tv_0$ where $v_0$ is the vector of club memberships the first year and $v_f$ is the vector of club memberships the second year, but again there is no problem in the question you have quoted. – Stella Biderman Sep 13 '17 at 18:45
  • I guess you have to find out the long-term distribution of the club-memberships. To get this, you have to solve a fix-point-equation of the form $Ax=x$ , so it boils down to find an eigenvector belonging to the eigenvalue $1$ with positive entries summing up to $1$ – Peter Sep 13 '17 at 18:45
  • @Manny In the first row, we do not have sum $1$, please check the entries! – Peter Sep 13 '17 at 18:47
  • 1
    @Peter Manny has a typo for Club A: It should read "$\frac{1}{10}$ switch to $B$." – John Wayland Bales Sep 13 '17 at 21:33

0 Answers0